The term ``biquad'' is short for ``bi-quadratic'', and is a common
name for a two-pole, two-zero digital filter. The
transfer function of the biquad can be defined as

(11.8)

where is called the gain of the biquad. Since both the
numerator and denominator of this transfer function are quadratic
polynomials in , the transfer function is said to be
``bi-quadratic'' in .

As derived in §10.1.3, for real second-order polynomials having
complex roots, it is often convenient to express the polynomial
coefficients in terms of the radius and angle of the
positive-frequency pole. For example, denoting the denominator
polynomial by
, we have

This representation is most often used for the denominator of the
biquad, and we think of as the resonance frequency (in
radians per sample--
, where is the resonance
frequency in Hz), and determines the ``Q'' of the resonance (see
§10.1.3). The numerator is less often represented in this way,
but when it is, we may think of the zero-angle as the
antiresonance frequency, and the zero-radius affects the
depth and width of the antiresonance (or notch).

As discussed in §10.6.4, a common setting for the zeros when
making a resonator is to place one at (dc) and the other at
(half the sampling rate), i.e., and
in Eq. (10.8) above
. This zero
placement normalizes the peak gain of the resonator if it is swept
using the parameter.

Using the shift theorem for z transforms, the difference
equation for the biquad can be written by inspection of the transfer
function as

where denotes the input signal sample at time , and
is the output signal. This is the form that is typically implemented
in software. It is essentially the direct-form I implementation. (To obtain the official
direct-form I structure, the overall gain must be not be pulled
out separately, resulting in feedforward coefficients
instead. See Chapter 9 for more about
filter implementation forms.)